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On the number of polynomials with small discriminants in the Euclidean and $$p$$-adic metrics. (English) Zbl 1275.11109
The main result of this paper is a lower bound for the number of polynomials with integer coefficients of bounded degree and height satisfying the property that their discriminant is ‘small’ and divisible by a large power of a fixed prime. The precise statement uses the following notation. Let $$P_n(Q)$$ denote the set of non-zero polynomials with integer coefficients of degree $$\leq n$$ and height $$\leq Q$$. The height of a polynomial is defined as the maximum of the absolute values of its coefficients. Let $$D(P)$$ denote the discriminant of a polynomial $$P$$. For $$v\geq0$$ let $\mathcal{P}_n(Q,v)=\{P\in P_n(Q):\;1\leq|D(P)|< Q^{2n-2-2v},\;|D(P)|_p<Q^{-2v}\},$ where $$|\cdot|_p$$ denotes the $$p$$-adic valuation. The authors prove that for $$n\geq3$$ and $$0\leq v<1/3$$ and all sufficiently large $$Q$$ one has that $\#\mathcal{P}_n(Q,v)\geq C\,Q^{n+1-4v}.$

##### MSC:
 11J83 Metric theory 11J54 Small fractional parts of polynomials and generalizations 11J61 Approximation in non-Archimedean valuations
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